For someone who is up for one night because of things on their mind, those phrases would probably mean essentially the same thing. People who regularly have sleep problems are likely to describe them more precisely, in which case the sentences could mean different things.
I couldn’t fall asleep at all last night.
The person never achieved sleep, and it implies that they were awake enough the whole night to be aware of the fact that they weren't falling asleep. To the comments about "at all" being ungrammatical, I would suggest that it implies a meaning of "at any time" rather than degree.
I couldn’t sleep at all last night.
I couldn’t get any sleep last night.
These are ambiguous, and the two sentences could apply to similar situations. They could have meanings ranging from literal to figurative.
Literal: they could mean that the person wasn't able to fall asleep, so they didn't get any sleep.
Perceived: they could mean the person perceived that they got no sleep, or almost no sleep. Without getting into the weeds of the physiology of sleep, there is a range of normal conditions in which the person perceives that they got little or no sleep but actual got more than they thought.
Figurative: the sentences could also mean just that it was a poor night's sleep that wasn't restful. The sentence could be figurative or an exaggeration.
These two sentences don't differentiate the various conditions they could describe and aren't reliably different from each other.